Chordal graphs are an important class of vertex decomposable graphs. Recall that a graph is called chordal, if all cycles of four or more vertices have a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle. Adam Van Tuyl, Rafael H. Villarreal in [1]} proved that all chordal graphs are (nonpure) shellable. Woodroofe in [2]} proved further that a chordal graph is vertex decomposable and later, generalized the idea to clutters in [3]}.
